On indefinite sums weighted by periodic sequences

TL;DR

This paper derives a formula to compute indefinite sums of sequences weighted by periodic sequences, expressing them in terms of sums over subsequences, and illustrates its application with examples.

Contribution

It introduces a new formula linking indefinite sums weighted by periodic sequences to sums over subsequences, enabling explicit calculations when available.

Findings

Provides a general formula for indefinite sums with periodic weights.

Enables explicit sum calculations using subsequence sums.

Includes illustrative examples demonstrating the formula's application.

Abstract

For any integer $q\geq 2$ we provide a formula to express indefinite sums of a sequence $(f(n))_{n\geq 0}$ weighted by $q$-periodic sequences in terms of indefinite sums of sequences $(f(qn+p))_{n\geq 0}$, where $p\in\{0,\ldots,q-1\}$. When explicit expressions for the latter sums are available, this formula immediately provides explicit expressions for the former sums. We also illustrate this formula through some examples.